ESPCI-MICHELIN Workshop, December, 2010

Theories of Activated Slow Relaxation, Aging and Mechanical Response of Glassy Polymer & Colloidal Materials

Ken Schweizer

University of Illinois @ Urbana-Champaign, USA

GOALS: “microscopic” statistical dynamical theory

unified approach: highly viscous liquids, glasses, “gels”

quiescent relaxation, aging, nonlinear viscoelasticity

connection to material-specific forces and structure

COWORKERS: Erica Saltzman (liquids) Dr. Kang Chen (glasses)

NSF-NIRT on glassy polymers with Mark Ediger, Jim Caruthers, Juan dePablo

Polymer Reviews: J.Phys.-Condensed Matter, 2009; Annual Rev.Condensed Matter Phys, 2010

Glassy Particle Suspensions

S(q)

q

r/D

0.0 0.5 1.0 1.5 2.0 2.5 3.0

g(r)

0

2

4

6

8

10

φ = 0.6

0.53

0.45g(r)

∗

∗

“Diverging” Relaxation Time, τα ?

r/σ

*

“cage”

* Tunable & Diverse Forces

* Nonspherical Colloid Shapes

* Soft & Deformable

Strong NONgaussian “Dynamic Heterogeneity” Effects

“Trapped” in a “dynamical precursor” world

BUT, still see Activated Hopping, rare event transport

Brownian time : τ0 =σ2 /DSE ~ 0.01-30 secparticle diameter ~ 100 nm -2 µm

WeeksWeitz

Hard Spheres

CAGE

intermittent trajectories

“kinetic vitrify” : τα ∼ 100 − 10,000 sec

STRUCTURE

Nonlinear Langevin Eqn Theory

!̂s(!r,t)="

!r#!ri(t)$

%&'

DERIVATION : KSS, JCP, 2005

Solid State View

r(t) = scalar displacement of a spherical particle

Saltzman & KSS JCP, 2003

Physical Ideas & Technical Approximations

* Key “slow variable” : Density Fluctuations

* Average over local packings: dynamical caging constraints via Effective forces S(q), g(r)

** Local Equilibrium Approx: relate 1 and 2 body dynamics

!"̂s(!r,t)

!t= D

s#2"̂

s(!r,t) + D

s#"̂

s(!r,t) d

!r '$ "̂(!r ',t)#V(

!r %!r ') + &

i#"̂

s(r,t)

!(2)!r,!r ';t( )

!(1)!r;t( )

" !g |!r#!r '|( )

Dynamic “closure”

Ds : dissipative, short time, “bare” processFormally:

ala DDFT

……a theory for single particle trajectories…seek Stochastic Eqn-of -Motion

!s"r(t)"t

= # ""rFeff(r(t))+ $(t)

white noise

!Feff(r) ="3ln(r)"1

3

d!q

(2# )3C2(q)$S(q)e

"q2r2 1+S"1(q)( )/6% & F

ideal+ F

cage

FAVORS: Delocalized Localized Liquid Solid

Nonlinear Langevin Eqn Theory …force balance in overdamped regime

Mean Square Caging Force from Structure

Time Local Displacement-Dependent “Field”

compete

Instantaneous Force on MOVING Particle due to Surrounding particles

“Dynamic Free Energy” =

FULL Dynamics ~ Sequence of locally complex in space-time noise-driven stochastic“events”

!!C(r)

!!C(r)

ρS

Dynamic Free Energy : Hard Spheres

φ = 0.3, 0.46,0.53,0.57,0.6

Displacement, r(t)

..

. .

*

*

rLOC

*

*

r(t)

“MCT transition” = Dynamic Crossover

Transient Localization & Hopping

φC ~ 0.432

Activated Transport

Hard Spheres

nMCT Freezing Kinetic Vitrify

φ

∗

Entropic Barrier

FB(φ)

~7-8 kBT

*“solid” glass

“normal” regime

Analytic Analysis

FB

∗∗

∗

rL

rBrrxn∗ Kramers theory: mean first passage time

!hop!0

=2" (#s /#0

)

K0KB

eFB ~ alpha time

@ cage scale

High Barrier Limit : Real Space Picture

!g2(" ) # F

B “contacts”

FB !"g2(# ) ! "

RCP$"( )

$2%&

“SOLID” only at RCP Jamming

Double Pole NOTWLF free volume

Impulsive Collisions

“mean square force” on moving particle

!s"r(t)"t

= # ""rFeff(r(t))+ $(t)

Noise-Driven Intermittency Trajectory Fluctuations

ALL single particle time correlations…. Heterogeneous Dynamics

Full Numerical Analysis

Alpha (cage scale) Relaxation

! * /!0 " exp F

B(#)( )

!

0.40 0.44 0.48 0.52 0.56

"*

0.1

1

10

100

q*

σ = 1 µm

τ α φ

5x104 s 0.57(14 hrs)

5 months 0.61 "glass"

Fs(q*,t)

approachRCP

! exp B / ("RCP

#")2( )2007 NLEprediction

Cipelletti et al,PRL, 2009extra 2 orders magnitude

MCT

~ double essential singularity

~ new Expts

!"

!0

φC,theory

MANY Heterogeneous Dynamics consequences

Diverse Singular forms *FIT* over ~ first 3 orders magnitude MCT critical power law, “free volume”, ……then ALL FAIL

barrier ~ kT

Hard Sphere

Janus Colloids

Biphasic MixtureGel-Glass competition

Depletion Gels

SOFT “MOLECULAR”

Colloids and Nanoparticles : A Zoology of Material Systems

Stars

Microgels

* Large Effects of Particle Softness & Overlap

* Coupled Activated Translation-Rotation Dynamics * Glass vs. Gel vs. Attractive Glass vs. Plastic Glass vs. Double Glass

Alpha (Segmental) Relaxation Map: Thermal Liquids

MCTCrossover Region

High Temperature Arrhenius

)(TLog!

10!

7!

42~ !

T

1

cT

1

gT

1

!"" )(~

cTT

AT

1

Ae!"

Tc : “dynamic crossover” ~ 10-7 s

Below Tg NONequilibrium

Arrhenius…why ?

Aging….how ?

Effect of Stress ?

Tg!150"500 K

Polymers

local nanometer scale physics

DEEPLY Supercooled NON-ArrheniusT–dependent barrier

NLE Theory of Molten Segmental Dynamics

“Liquid of Gaussian Segments”

σ!!"jr

Displacement

!sdr(t)

dt= " d

drFeff(r(t)) + #(t)= 0

COARSE GRAIN to segment nm scale

! = C"l

S(q~0) ! S0

= " kBT #

T$ (%")2 "&2 $ &A+B

T

'()

*+,

&2

KEY : Dimensionless Compressibility or inverse Bulk Modulus ~ amplitude of nm scale density fluctuations

measurable via Scattering or Thermo …key to dynamical predictive power

r(t)

Local Angstrom Cage scale “chemistry” Averaged Over

T, P, N, chemistry dependentCohesive Energy Density key

Intra-Segment scale fast Arrhenius Segment-Segment forces

simplicity

emerges

S(q)“fast”

!(T ) " 1

#$3S03/2

SINGLE parameter quantifies Localizing Forces Feff (r)

λ > λc

λ < λc = 8.3

*

r /σincreases as cool or pressurize

Mean Segmental Hopping (alpha) Time

!"(T ) = !ARR

(T )eac#FB(T )

smooth crossover at Tc Arrhenius to Supercooled

σ!!"jr

* POLYMER Stiffness Effect : ac Cooperative Segment Hopping

DYNAMIC SEGMENT Kuhn or Persistence Length …MAP ac! " l

K=C

#l OR lp = C#+1( ) / 2$ ac ~1%9

NON-Arrenhius NON-analytic in T NON-singular at T > 0

a priori , sensible ~ 280 K, PDMS 460 K, PolyEtherImide

FB! "#"

c( )1.4

Barrier

Kramer’s

Inspired

Deeply Supercooled a-PMMA Melt

“Glass”

Expt Tg ! 378K if ac ! 5,6 ...predict m ~ 120!130 ~ expt

m !

d(log"#

)

d(Tg / T )Tg+

*

CH3CH2-C-

|

O=C-O-CH3|

Dynamic Fragility

~ range of MOST polymers

m ~ 55 --> 140

General Message : to leading order, fragility determined by backbone stiffness

Can FIT calculations VERY well to singular VFT or WLF …..but not true in NLE !

!" # exp

B

T$T0

%

&'

(

)*

m!16+40 a

c

•

1 new material parameter : b SAXS Expts : b ~ 0.4 - 0.7

SAXS: S(q ~ 0) = S0

S0(T < T

g) ! bS

0(T

g) +

T

Tg

(1" b)S0(T

g)

melt

glass

T < Tg:Collective Density Fluctuations in NONequilibrium GLASS

Basic Landscape Concepts

Frozen (δρ)2Inherent structures

Equilibrated ~ Vibrations

Tg

*

T

Teff

Tg

Alba-Simionesco et alMacromolecules, 2008

Confining FORCES:same physical picture as ABOVE Tg

Age

linear

“quenched”

Crossover to Arrhenius Relaxation (modestly) below Tg

b = 0 0.33 0.5 0.67 0.75

Activation Energy RATIO R ~ 3 - 6

PHYSICS : NONequilibrium effectEXPTS : diffusion,

dielectric, mechanical…. R = E

A

+/ E

A

!" 3 ! 6

disordered solid (!")2.... S

0(T )

Morefrozen (δρ)2

Arrhenius for all b Weaker Constraints

*

* ignore physical AGING….. “rapid quench experiment”

Equilibrium

Age of Universe !

(quenched) Glass Linear Elasticity

Consistent with Mechanical Definition of Glass via E’(T)

Young’s Modulus + Stress-Stress TCF + Green-Kubo +Maxwell

E ' ! 2.8

60!2dq0

"# q4 $lnS(q)

$q%

&''

(

)**

2exp +q2r

loc2 /3S(q)%

&'()* * r

loc

1.5 GPa @ Tg

~ linear

σ ~ (lp , lK ) ~ (0.75, 1.35) nm

a-PMMA

Mulliken & Boyce

Feff(r)

fit

MUCH stronger growth in liquid

Enthalpy, Density,… Modulus Yield Stress…..

Glass is NONequilibrium state…..…. Time-Dependent Properties more “solid-like” as equilibrate via local activated α -process

Sigmoidal Response McKenna, JPCM, 2003 equilibrium

!"! t

age

µ

exponent : µ(T) grows with cooling

plastics used “close” to TgQuiescent Physical Aging

Relaxation Time

Age Logarithmically ~ Log{tage}

“Down-Jump” Expt : quench below Tg and wait

ThermoMechanical Properties

Theory of Physical Aging

aging

S

0(t) ! " (#$)2 ! % F

B(t) & % '( (t), G ', '

y&

dS0

(t)

dt= !

S0

(t)! S0,l

"#

(t;S0

(t))

S0, l

: final equilibrium value

S0, g

: initial nonequilibrium value

PMMA

as time passes, equilibrate via Activated Hopping

PRL, 2007PRE,2008

NON-linear & Self-Consistent… No adjustable parameters

4% differ @ Tg-T = 8 K

Tg

“down jump” expt

ANSATZ

S0(t) = S

0 , l+ S

0(t = 0) ! S

0 , l( )exp ! dt ' "

#

!1

(t)

0

t

$%&'

()*

! logarithmic, slope " as cool

Density fluctuations, Cohesive energy, Modulus,…..

Aging Predictions for a-PMMA

log-log plotnormalized time, τα(0)

DOWN-JUMP :

Good Power laws

!" ! tage

µ

UP-JUMP :365 K --> 370 K VASTLY different: long incubation..then not power law

Correct Asymmetry

Tg = 378 K

S0,g

! S0

(t )

S0,g

! S0,l

Down Jump Power Law : T-dependent Aging Exponent

!" ! tage

µ(T )

Deeper Quench

Higher exponent

Faster AgingPS

NONuniversal Material Aspects

Struik “downturn”

Mechanical Behavior of Amorphous Polymer Plastics

1) Elastic solid

2) Strain Softening + “overshoot yield peak”

3) Dynamic Yielding segment scale “plastic flow”

Deformation-induced de-vitrification Tg(stress)

4) Strain Hardening uniquely polymeric…chain deformation

Physical Aging

Stress-Driven “Rejuvenation”…mechanical disordering…new steady state

coupled effects on mechanics

Stress - Strain

1

2

3

4

~ 5-10%

Fix T & Rate pre-aging time

* Flow Stress

* Simulations (Rottler, Robbins, dePablo…) * Experiments (Ediger,…)

Tight Correlation of isotropic SEGMENT Dynamics and Bulk Mechanical Properties

*Strong T & Strain Rate Dependence …..”Activated Process”

0th order Incorporation of Stress in NLE Theory

τ = Applied Stress

Reduces Modulus & Lowers Barrier

Accelerates Relaxation

External force on particle

r

F(r;!)=F(r;! =0)"##2! r

τ increases

!hop

!0

=2" g(# )

K0(! ) K

B(! )

eFB(! )

G '(! )= 1

60"2dq0

#$ q4

% lnS(q)%q

&

'(

)

*+

2

e,q2r

LOC2 (! )/3S(q)

ala isotropic Eyring @ “instantaneous dynamical variable” level

Mechanical Work

*

Stress “tilts landscape”

Kobelev+KSS PRE 2005

FB

(! ) " FB

(0) 1#(! /!y,abs

)$%&

'()

5/2

(transient) Glassy Modulus

rLOC

BELOW Hardening regime :little change in conformation, packing,…

Barrier Softening & “Stress-Induced De-vitrification”

Stress

Melting

…glass flows !

Input to Constitutive Eqn

1000 s

Barriera-PMMA GLASS

Tg = 378 K

T < Tg

* Neglect Aging “rapid quench”

*NEGLECT Mechanical Induced Structural Disorder

S0! f ( !" , stress, t) soley due to

“Landscape Tilting”

Generalized Maxwell Constitutive Eqn

0( '' )';( ) ( )

t

sE t t tresst d tt !" #= $ !

Ansatz : Nonlinear Boltzman superposition

“stretched” Maxwell Model:

E(t ! t ';stress) = E '(" (t '))exp ! dt ''"#

!1 " (t '')$% &'t '

t

()*

+,

! (t) = dt ' E '(! (t '))e

" dt ''!#"1 ! (t '')$

%&'t '

t

(!) (t ')

0

t

(

• Constitutive Equation

/ 2(1 ) 2.5 3E G !" " = + #!

“Effective Time” form

APPLY : constant strain rate

Nonlinear Self-Consistent Eqn for Stress(t)

Strain ! " = !" t

MacromoleculesJCP, 2008

(minimal) Input Physics : Stress-dependent Elastic Modulus & α−Relax Time

Comparison with a-PMMA Expts (rapid quench)

!! = 0.001

0.005

0.0001 sec"1

Tg-T = 97 K79674227

Hasan, Boyce et al. JPSPP, 1993

Lee & Swallowe, JMS,2006

Dynamic FLOW STRESS

τy(T) ~ linear in temperature

Yield STRAIN ~ 5-10%

Alpha Time AT Yield “short”….glass melting

FIT

100 s

FIT

yield

“Deformation Thinning” in Flowing GLASS per Complex Fluid

!! "# (0)( )c<<1

Master Curvemaster curve …per Ediger Expts & Sims

Riggleman, Schweizer, dePablo, Macromolecules 2008 ISOTROPIC acceleration TOO !

*

…“thinning” onset at

Plastic Flow regime “locally equilibrated” liquid-like nanodomains

!! "#,yield$%

&'<<1

More at higher T / Lower rates

! !" #0.9±0.1

Log(Pe)=

power law

a-PMMAMontes et al, Macromolecules, 2008

“Hardening Modulus” GR

Recent Experiments &Simulations (Hoy & Robbins; Lyulin,…)

• ala Ideal Entropic Intrachain Rubberλ−Dependence

**BUT**

• Magnitude > 100 times rubber Modulus

• Increases with Cooling…NOT entropic !

• Strain Rate dependent

• Correlates with Yield Stress

Suggests local dissipative process in glass relevant…SEGMENTAL RELAXATION

Classic Model qualitatively WRONG

Strain Hardening

λ

Theory : Suppressed Density Fluctuation Origin

Uniaxial Deformation --> Anisotropic single Chain Conformation

induce anisotropic INTERchain correlations….POST-Yield ”local equilibration”

PRISM THEORY : reduces density fluctuation amplitude, S0(λ) ….origin of deformed rubber network has INCREASED Tg

Enhances dynamical constraints, Barrier --> Slows Relaxation, Raises E’

λ

0 1 2 3 4 50.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.0, 501.0, 1001.0, 2500.5, 1002.0, 100

λ

�

S(!)S(1)

Anisotropic PRISMStrained rubber networks& melts Oyerokun & KSS, 2004

S0(!)

S0(1)

=1

1+3S

0(1)

16

"r

X

!2+2!#1

3#1

$

%&

'

()

*

large stress upswing…..”Strain Hardening” due to Glassy Physics

Length Scale of Affine Deformation ? (effective crosslinks) scale beyond which glass = “solid”

rX ~ 4 nm ~ Scattering & Mechanical PMMA Expts , Tg-T ~ 20 K….NX ~ 16 Montes et al, Macromolecules, 2008

PRL, 2009

NONmonotonic Alpha Relaxation Time

Enhancement MODEST (Ediger Experiments)

…..BUT “big enough”

Tension

Compression

CoolIncrease Strain Rate

Vary Tg-T = 90, 70, 50, 30, 10 @ rate = 0.001 s-1

Vary RATE = 10-5 - 0.1 sec-1

@ fixed Tg-T = 50

NX=16

Extract Hardening Modulus g(!) = !

2" !

"1

Hardening

Correct “rubber-like” form

Dynamic Yield Stress

!y(T , !" )

GR

Yield

Many Open Issues

GR ~ 100 Ge ~ even close to Tg

~ Linear with T

~ Log{strain rate}

correlates with Yield Stress

absolute numbers reasonable (factor 2-4)

Tension

Yield Stress@ 10-3 s-1

Theory Results : NON-Entropic Hardening Modulus

Predict non-affine length scale ?….rate,T dependence ? micro- vs. macro- stretch ?

role of entanglements, Ne ? role of chain scale dynamics ?

tension vs compression asymmetry ?

Compress

Rob HoyMark RobbinsCorey O’HernRob Riggleman

* NEED: Exptl Probe Density Fluctuations in Hardening Regime….SANS, USAXS ?

strain rate 10-1 Hz 10-3

10-5

Other Phenomena & NLE Polymer Glass Theory

Creep ; Creep RecoveryStrain Softening, Yield Peak

due to COUPLED Mechanical Rejuvenation & Physical (pre) Aging JCP, 2008

PRE, 2010+ in progress

Lee, Ediger, et al, Science, 2009

Nature of Nonequilibrium Steady State ? aging erasure…..Viscosity Bifurcation: aging under stress Aging wins vs. Rejevenation wins ?

S0= f ( !! , stress, tpre"age,t)

MANY Open Questions & Theoretical Challenges

Supercooled Melts : * Decoupling of Segment Scale & Chain scale relaxation

recent new idea : Alexei Sokolov & KSS, PRL, 2009

* Dynamic Heterogeneity effects, KWW relaxation, Chain modes….

Polymer Glass : * Kovacs aging “memory effect”…multiple temperature jumps

* Tensorial deformations : tension vs. compression vs. shear…..

* Role of Dynamic Heterogeniety in “solid state” mechanics ?

Polymer Nanocomposites : marriage of the colloidal and polymeric worlds !glasses and gels relevant

Colloidal Glasses & Gels : nature of Aging, Mechanical Rejuvenation,…..

Beyond NLE : 2-particle dynamic correlations Daniel Sussman & KSS, JCP, submitted, November, 2010